"Cis" and "trans" are different types of a person's.... what?

You might not believe in the social construct at birth, but the social construct believes in you. Children are treated differently based on assigned gender from birth.

No, gender is a social construct and the doctor is assigning a gender to you when you are born based on what he sees as your genital configuration. This is then used to determine nearly everything about you through the social framework of gender.

What colors you're allowed to like, what games you can play, what names you can have, what words are acceptable to refer to you with, who you're allowed to be friends with, what foods your supposed to like, what clothes you're allowed to wear, how people should speak to you, how people should praise you, how people should scold you, whether or not misogyny should be applied to you, and so on and so forth.

Those things are determined based on the gender you are assigned at birth. Those things are enforced across all society at all social levels and in all settings. Parents are the first people to enforce gender onto their children, intentionally or not. Then every single other adult and child they meet or interact with throughout their childhood will continue to enforce gender upon them until they themselves become adults and repeat the cycle with their own kids. Media perpetuates gender, government laws enforce gender, education systems are filled with people who systematically enforce gender upon children.

Thats what we mean when we say gender is a social construct. And you're assigned one at birth.

Sex is also a social construct btw.

I understand what they are, I'm asking if there is a name for the category of characteristic that they both belong to.

You're not getting an answer to your question because the question, as stated, is incomprehensible. You're asking for a "category of characteristic" that a pair of antonym adjectives "belong to"? That doesn't make sense. They apply to a whole host of characteristics, because they're not describing a specific characteristic, but how a characteristic relates to the whole. Just like "homo" and "hetero"; homozygous, heterogenous, homocystine, and heterophony are all words that use the "homo" or "hetero" prefix to describe how those words relate to other concepts in their category. It's the same with "cis" and "trans". The prefixes don't "belong" to a category of characteristics, they explicitly exist outside of the characteristics of the words their modifying.

That's the best I can do with the way you've chosen to phrase your question, and I admit it's a reach, but your question is gibberish.

"Relationships between gender identity and birth sex."

There is no such category. Being cisgender or being transgender describes the relationship between 2 variables. The first being your assigned gender. The second being your gender identity. Cisgender means there is an equivalence of those 2 variables. Transgender means there is not an equivalence of those 2 variables.

The reason we use the term trans which means roughly "other side" to describe this is because you cannot know you are transgender at birth. Your gender identity is assumed to be cisgender, it is assumed to be the same as the gender you are assigned. So when you reveal your gender identity to in fact be something different you are moving to another side of gender. At least in literal usage of the terms cis and trans.